The Design of Optimal Third-Order Bang-Bang Systems Using Delayed Switching

The time optimal performance of bang-bang control systems can be achieved by properly timing the relay operations, two switchings being required for the optimal response of third-order systems. The movement of the state point is observed in the two-dimensional projections of the phase space. If the first switching is delayed after the state point has crossed the optimum curves in both planes, the second switching occurs at the optimum switching boundary. The coordinate axis and the optimum boundary divide the phase space into sections described by logic variables, from which the controller has been developed. A function generator determines the projections of the optimum switching curve, and a delay circuit is designed to delay the first switching operation.